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Chapter 5: Problem 65

Describe a situation in which a right triangle and a trigonometric function are used to measure a height or distance that would otherwise be inconvenient or impossible to measure.

Short Answer

Expert verified
The height of the lighthouse can be measured using the distance the ship moved parallel to the shore and the angle of elevation of the lighthouse from the ship, using the formula: Height = d * tan(\(\theta\) ) where d is the distance moved by the ship and \(\theta\) is the angle of elevation.

Step by step solution

01

Describe the Scenario

Consider a scenario where there's a lighthouse on the seashore and a ship at sea, wanting to calculate the height of the lighthouse. It would be inconvenient or in some cases impossible to measure this height directly. So, trigonometry can be used here.
02

Detail the Process

The ship will move a certain distance parallel to the shoreline, let's say 'd', forming a right triangle with the lighthouse and its initial and final position. The sailors on the ship measure the angle of elevation of the lighthouse from the ship at its final position, let's say this angle is '\(\theta\)'. Now, using the trigonometric function 'tan', which is opposite side/adjacent side, in this case, the height of the lighthouse over the distance 'd'.
03

Formulate the Calculation

The height of the lighthouse can then be calculated using the formula: Height = d * tan(\(\theta\) ).

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Most popular questions from this chapter

Exercises \(127-129\) will help you prepare for the material covered in the next section. In each exercise, let \(\theta\) be an acute angle in a right triangle, as shown in the figure. These exercises require the use of the Pythagorean Theorem. If \(a=1\) and \(b=1,\) find the ratio of the length of the side opposite \(\theta\) to the length of the hypotenuse. Simplify the ratio by rationalizing the denominator.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using the equation \(y=A \sin B x,\) if 1 replace either \(A\) or \(B\) with its opposite, the graph of the resulting equation is a reflection of the graph of the original equation about the \(x\) -axis.

Describe an angle in standard position.

Explain how to find the radian measure of a central angle.

Will help you prepare for the material covered in the next section. a. Graph \(y=-3 \cos \frac{x}{2}\) for \(-\pi \leq x \leq 5 \pi\) b. Consider the reciprocal function of \(y=-3 \cos \frac{x}{2}\) namely, \(y=-3 \sec \frac{k}{2} .\) What does your graph from part (a) indicate about this reciprocal function for \(x=-\pi, \pi, 3 \pi,\) and \(5 \pi ?\)
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